Question
Show that $\big|\vec{a}|\ \vec{b}+\big|\vec{b}\big|\ \vec{a}$ is perpendicular to $\big|\vec{a}|\ \vec{b}-\big|\vec{b}\big|\ \vec{a},$ for any two nonzero vectors $\vec{a}\ \text{and}\ \vec{b}.$
$=l^2\Big|\vec{b}\Big|^2-lm\vec{a}.\vec{b}+lm\vec{a}.\vec{b}-m^2\big|\vec{a}|^2$
$=l^2\Big|\vec{b}\Big|^2-m^2\big|\vec{a}|^2$
$\text{Putting},\ l=\big|\vec{a}\big|\ \text{and}\ m=\Big|\vec{b}\Big|,$$=\big|\vec{a}\big|^2\big|\vec{b}\big|^2-\big|\vec{b}\big|^2\cdot|\vec{a}|^2$
$\Rightarrow\ \ \ \ \ \ \big|\vec{c}\big|.\Big|\vec{d}\Big|=0$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\int2\text{x}^3\text{e}^{\text{x}^{2}}\text{dx}$